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Placeholder Substructures: The Road from NKS to Small-World, Scale-Free Networks is Paved with Zero-Divisors (and a “New Kind of Number Theory”) Robert de Marrais NKS 2006 Wolfram Science Conference – June 17. Keep it simple …. … and keep it stupid!.

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Placeholder Substructures:The Road from NKS to Small-World, Scale-Free Networks is Paved with Zero-Divisors (and a “New Kind of Number Theory”) Robert de MarraisNKS 2006 WolframScience Conference – June 17

the argument simply put
Complex (scale-free, small-worlds) networks are best comprehended as a side-effect of NKN (a new kind of Number Theory) which is …

Based not on primes (Quantity), but bit-strings (Position).

The role of primes is taken by powers of 2 (irreducible bits in “prime positions,” instead of “prime numbers”)

All integers > 8 and not powers of 2 have bit-strings which can each uniquely represent a “meta-fractal,” which we’ll call a “SKY”

Integers thus construed are called “strut-constants,” of ensembles of “Zero-Divisors.”

The argument, simply put:
now for the stupid part
Now for the “stupid” part:
  • Zero-divisors (ZD’s) are to singularities (nested, hierarchical, invisible, yet unfoldable by morphogenesis) …
  • … what cycles of transformations are to groups (heat into steam into electricity into keeping this slide-show running, say…).
  • As we trace edges of a zero-divisor ensemble, we keep reverting not to an “identity,” but to “invisibility” (“Nobody here but us chickens”): for triangle of ZD nodes ABC, A*B = B*C = C*A = 0.
  • “I see your point” means a whole argument indicated by a pronoun: a Zero “place-holder” with indefinitely large (and likely nested) substructure. ( “Point well taken!” )
  • An ensemble, that is, of Zero-Divisors, whose “atom” flies under the stupid name of “Box-Kite” (which flies in meta-fractal “Skies”)
the secret of our success
The secret of our success?
  • Starting with N=4, ZD’s emerge in 16-D; the simplest Sky in which Box-Kites fly in (infinite-dimensional) fractals emerges in 32-D.
  • Hurwitz’s 1899 proof showed that generalizations of the Reals, to Imaginaries, Quaternions, then Octonions, by the Cayley-Dickson Process of dimension-doubling (CDP), inevitably led to Zero-Divisors (in the 16-D Sedenions)
  • Fields no longer could be defined, and metrics broke. (Oh my!)
  • So (as with the “monsters” of analysis, turned into fractal “pets” by Mandelbrot), everybody ran away screaming, and never even gave a name to the 32-D CDP numbers
  • But these 32-D “Pathions” (as in “pathological”) are where meta-fractal skies begin to open up! (Moral: if you want to fly a box-kite, run toward turbulence! Point your guitar into the amplifier, Eddie!)
vents sails and box kites

Vents, Sails, and Box-Kites

This is an (octahedral) Box-Kite: its 8 triangles comprise 4 Sails (shaded), made of mylar maybe, and 4 Vents through which the wind blows.

Tracing an edge along a Sail multiplies the 2 ZD’s at its ends, making zero.

Only ZD’s at opposite ends of a Strut (one of the 3 wooden or plastic dowels giving the Box-Kite structure) do NOT zero-divide each other.

vents sails and box kites10

Vents, Sails, and Box-Kites

The strut constant (S) is the “missing Octonion”: in the 16-D Sedenions, where Box-Kites first show up, the vertices each take 2 integers, L less than the CDP “generator” (G) of the Sedenions from the Octonions (23 = 8), and U greater than it (and <> G + L).

There being but 6 vertices, one Octonion must go AWOL, in one of 7 ways. Hence, there are 7 Box-Kites in the Sedenions.

But 7 * 6 = 42 Assessors (the planes whose diagonals are ZD’s!)

vents sails and box kites11

Vents, Sails, and Box-Kites

It’s not obvious that being missing makes it important, but one of the great surprises is the fundamental role the AWOL Octonion, or strut constant, plays.

Along all 3 struts, the XOR of the opposite terms’ low-index numbers = S(which is why, graphically, you can’t trace a path for “making zero” between them!). Also, given the low-index term L at a vertex, its high-index partner = G + (L xor S):

S and G, in other words, determine everything else!

a different view with numbers too

A different view, with numbers too!

Arbitrarily label the vertices of one Sail A, B, C (the “Zigzag”).

Label the vertices of its strut-opposite Vent F, E, D respectively.

The L-indices of each Sail form an Octonion triple, or

Q-copy, since such triples are isomorphic to the Quaternions.

But the L-index at one vertex also makes a Q-copy with the

H-indices of its “Sailing partners.”

Using lower- and upper-case letters, we can write, e.g.,

(a,b,c); (a,B,C); (A,b,C); (A,B,c ) for the Zigzag’s Q-copies.

And similarly, for the other 3 “Trefoil” Sails.

a different view with numbers too13

A different view, with numbers too!

Note the edges of the Zigzag and the Vent opposite it are red, while the other 6 edges are blue. If the edge is red, then the ZD’s joined by it “make zero” by multiplying ‘/’ with ‘\’: for S=1,

in the Zigzag Sail ABC, the first product of its 6-cyle {/ \ / \ / \} is

(i3 + i10)*(i6 – i15) = (i3 – i10)*(i6 + i15) = A*B = {+ C – C} = 0

For a blue edge, ‘/’*’/’ or ‘\’*’\’ make 0 instead: again for S=1,

in Trefoil Sail ADE, the first product of its 6-cycle { / / / \ \ \ } is

(i3 + i10)*(i4 + i13) = (i3 – i10)*(i4 – i13) = A*D = {+ E – E} = 0

a different view with numbers too14

A different view, with numbers too!

One surprisingly deep aspect among many in this simple structure: the route to fractals is already in evidence!

The 4 Q-copies in a Sail split into 1 “pure” Octonion triple and 3 “mixed” triples of 1 Octonion + 2 Sedenions; the 4 Sails also split: into one with 3 “red” edges, and 3 with 1 “red,” 2 “blue.”

Implication: the Box-Kite’s structure can graph the substructure of a Sail’s Q-copies – which is not an empty execise! Why?

Take the Zigzag’s (A,a); (B,b); (C,c) Assessors and imagine them agitated or “boiled” until they split apart. Send L and U terms to strut-opposite positions, then let them “catch” higher 32-D terms, with a higher-order G=32 instead of 16. We are now in the Pathions – the on-ramp to the Metafractal Highway!

strut opposites and semiotic squares
Strut Opposites and Semiotic Squares

René Thom’s disciple, Jean Petitot, has been translating the structures of literary and mythic theory – Algirdas Greimas’ “Semiotic Square,” Lévi-Strauss’ “Canonical Law of Myth” – into Catastrophe Theory models; here, we translate these into Box-Kite strut-opposite logic: ZD “representation theory” as semiotics.

From here, we’re off to Chaos! We’ve just one stop left: another “representation” of Box-Kite dynamics – the ZD “multiplication table” called an ET (for “Emanation Table”)

the simplest sedenion emanation tables

The Simplest (Sedenion) Emanation Tables

For S=1 Box-Kite, put L-indices of the 6 vertices as labels of Rows and Columns of a ZD “multiplication table,” entering them in left-right (top-down) order, with smallest first, and its strut-opposite in the mirror-opposite slot: 2 xor 3 = 4 xor 5 = 6 xor 7 = 1 = S.

If R and C don’t mutually zero-divide, leave cell (R,C) blank.

Otherwise, enter the L-index of their emanation (the 3rd Assessor in their common Sail). (Oh, yeah: ignore the minus signs.)

cantor dust curdled

Cantor Dust, “Curdled”

“The very same properties that cause Cantor discontinua to be viewed as pathological are indispensable in a model of intermittency.”

- Benoit Mandelbrot

georg cousin moritz says hello

Georg, Cousin Moritz says hello!

“Given Cantor’s Rosicrucian theology and the proximity of his cousin Moritz Cantor – at that time a leading expert in the geometry of Egyptian art (Cantor 1880) – it may be that Georg Cantor saw the ancient Egyptian representation of the lotus creation myth and derived inspiration from this African fractal for the Cantor set.”

– Ron Eglash, African Fractals

instant fractals course 1 slide s worth
Instant Fractals Course (1 slide’s worth)

Ethiopian Processional Crosses (L-to-R => seed + 3 iterations)

limit case ces ro double sweep

Limit-case:Cesàro Double-Sweep

One of the simplest (and least efficient!) plane-filling fractals, its white-space complement is clearly

approached by the S=15 meta-fractal Sky!

s 15 sky emerges in 32 d pathions

S=15 Sky emerges in 32-D Pathions

6 Sedenion Assessor-dyads of S = 7 Box-Kite SPLIT UP:

L (index < 8) and U (index > 8) units all become L-units

(index < 16 = new G) in Pathion 3-BK ensemble with S = 15 (= 8+7 ),

along with prior G (=8) & S (=7), which capture U-units (index > G)

from ambient turbulence, resulting in 14 Pathion Assessors

slide60

2nd Nested “Sky-Box” emerges in 64-D:30 “blue-sky” border cells, one per each newAssessor in the 26-ions (“Chingons”)

Prior iteration’s row and column LABELS become blue-sky CELLS! (with label-to-cell mirror-reversal in “strut-opposite” boundary walls). This iteration’s 30 ( = 2[N-1] - 2, N = 6) row and column LABELS will

in turn become blue-sky CELLS in the next, 62-cell-edged, iteration:

scale free boogie woogie

Scale-free Boogie-Woogie

(CL)AIM: Sky meta-fractal gridworks are dynamic.

Like Mondrian’s capturing of New York City’s

creative bustle in canvas-fixed oils, “complex net-works” can be modeled by interlocking emanation tables’ “small-worlds” and “scale-free” givens!

sky pilot
SKY PILOT!

A SKY is made of many layers, like the “sheets” of a Riemann surface in complex analysis.

Navigate between sky-box-bounded spreadsheets like David Niven in Around the World in 80 Days:

sky pilot64
SKY PILOT!

Go up higher? Push G a bit to the left! (David drops a

sand-bag over the edge, to same effect.)

Drop down lower? Shift G to the right! (David vents

hot air by pulling a cord: same thing.)

sky pilot65
SKY PILOT!

Remember: if you know S and G, the L-index is all you see in the ET,

and all you need to see: the U-index is always just G + (L xor S)

… with G an “infinite constant” in the “meta-fractal” limit-case!

(Up, up and away!)

cooking with r cip s
Cooking with Récipés
  • Strut-Constant-Emanated Number Theory (SCENT) is the basis of R, C, P’s: simple formulas specifying the relations between Row and Column labels, and their XOR Products housed in the spreadsheet-like cells of Emanation Tables (ET’s).
  • For all S > 8 and not a power of 2, there exists a unique meta-fractal or “Sky,” whose ET has a simple algorithm.
  • For any cell, consider the bit-representation of S; the cell is filled or empty (shows or hides P) depending upon a series of “bits to the left” tests, starting with the highest, and stopping at the lowest (if the 3 rightmost bits > 0) or next-to-lowest (if S = multiple of 8 and not a power of 2).
  • We’ll build to the general case from the simplest ones.
the simplest flip book r cip
The Simplest “Flip Book” Récipé
  • Pathion “Flip-Books” obey this formula: R|C|P = S|0 mod 8
  • This (and all more complex formulas) always implicitly assume a “zeroth rule”: All long-diagonal cells are empty
  • (The ‘\’ has all R=C, so all P = R xor C = 0, and i0 is the Real unit!)
  • (The ‘/’ has R and C as strut-opposites, which also can’t mutually zero-divide, since P = R xor C = S … and S is suppressed in ETs).
  • The Flip-Book Formula then says: after blanking out diagonals, fill all cells with Row or Column or cell-value Product equal to (S – 8), or 8 itself, then leave the rest empty.
  • For 8 < S < 16, the 1st and last rows and columns start off defining a square, then move toward each other (with P’s forming diagonals joining the central cross when S = 15), as S is incremented.
add parentheses get balloon rides
Add Parentheses, Get “Balloon Rides”
  • For 8 < S < 16 in the 64-D Chingons, recursive structure emerges, based on the Pathion flip-book: R and C labels of the latter become bordering cell values along left and top edges (and mirrored on the other sides) of an exact copy of the Pathion ET.
  • With each dimension doubling, the ET’s edge tends toward doubling too (= 2[N-1] – 2, for 2N-ions); the count of filled rows and columns tends likewise ( #R = #C = 2 * {2[N-4] – 1} ), with prior iterations’ labels mapped to current cellvalues along all “skybox” edges.
  • And always, the formula is just what we saw above, but with parentheses added: (R|C|P = S|0 ) mod 8
  • That is: all multiples of 8 < G, and S modulo each, now appear: for S=15, this means {8,7} plus {16, 23; 24, 31} for 26-ions (G=32); for 27-ions (G=64), these plus {32, 39; 40, 47; 48, 55; 56, 63}; etc.
next add rules one per hi bit
Next, Add Rules (one per hi-bit)
  • For Chingons with S > 16, we get new behavior, best appreciated by contrasting the cases S = 24 and S = 25.
  • As a bit-string, 24 = 11000, with 2 hi-bits; but, with no low-bits, the 8-bit is treated as one, yielding one fill rule (and an empty central box): ( R|C|P = 8 | 0 ) mod 16.
  • But 25 = 11001, with 2 effective hi-bits, hence a fill rule (off the 16-bit) and a hide rule (off the 8-bit). As with cooking a stew, once you toss something in, you can’t take it out: a cell, once a rule has been applied to it, can not have its hide/fill status changed by later rules.
  • 24 ends with a fill rule, so all cells not ruled upon are left empty; but 25 ends with a hide rule, so all untouched cells are filled.
canonical r cip s
If S, asstring, has hi-bits b1,b2,…,bk in L-to-R positions from 2H to 2L (L > 3):

base a fill rule on all “ON” bits bi where i = odd;

base a hide rule on all “ON” bits bj where j = even.

If the last rule is hide, then fill all cells untouched by a rule;

if the last rule is fill, then hide all cells untouched by a rule.

For any hi-bit 2A, the rule has form ( R|C|P = (S|0) ) mod 2A, with all nominated cells filled or hidden according to case.

To see recipes at work, the simplest abutment of 2-rule and 3-rule S values ( S = 56 and 57, respectively, in the 128-D 27-ions, or “Routions”) are illustrated in stepwise detail in what follows.

Canonical Récipés